Finally, for any \(v\) -stack \(X\) , write \(D_{\mathop {\mathrm {\acute{e}t}}}(X,\mathbf {Z}_\ell ) \subset D_{\ell -\text{comp}}(X_v,\mathbf {Z}_\ell )\)  as the full subcategory derived \(\ell \) -complete objects in \(D(X_v,\mathbf {Z}_\ell )\)  whose mod \(\ell \) -reduction lies in the subcategory \(D_{\mathop {\mathrm {\acute{e}t}}}(X,\mathbf {Z}/\ell )\)  mentioned above ([1]}). If \(X\)  is a locally spatial diamond, then derived \(\ell \) -completeness as well as the previous remark on left-completeness give an equivalence
\(\mathcal {D}_{\mathop {\mathrm {\acute{e}t}}}(X,\mathbf {Z}_\ell ) = \lim _n \mathcal {D}_{\mathop {\mathrm {\acute{e}t}}}(X,\mathbf {Z}/\ell ^n) \simeq \lim _n \widehat{\mathcal {D}}(X_{\mathop {\mathrm {\acute{e}t}}},\mathbf {Z}/\ell ^n)\) 
